# Flat yet finite Universe

I would like to ask you this question:

Let's assume that the universe is perfectly flat and big bang happened as described by our theories some 14 billion years ago. Shouldn´t, therefore, the radius of the universe still be finite, even if the geometry of the universe is flat? Even if inflation happened and even if the universe itself could expand faster than light without violating relativity, both speeds were finite, shouldn´t, therefore, the universe be also finite?

I mean the universe couldn´t start finite time ago, expanded with finite speed and yet be infinite, that is a contradiction, so what gives?

• Why do you think universe was finite at big bang? If we assume that universe is flat and infinite, that'd mean it has been like this forever (i.e it's infinite). And at the big bang itself, we will encounter an ambiguity, something like $0* \infty$ which is not necessarily finite – Paradoxy Jul 29 '19 at 0:07

First let's note that the notion of perfect spatial flatness applied to our universe is a mathematical property which one would not expect the physical universe to show. However we can explore what sort of universe could have that property and still fit with general relativity.

Here are the possibilities:

1. The universe is finite and spacetime has a boundary of some sort in spatial directions as well as past temporal direction.

2. The universe is spatially finite and unbounded.

3. The universe is spatially infinite and always has been.

Note there is not a 4th option where the universe is spatially infinite now but was not in the past. You are right to suspect that would not be possible. However, there are some subtleties to do with the relativity of simultaneity, such that depending on how you define what events to call simultaneous, the same situation in spacetime can be described either as finite volume with regions of infinite density, or as infinite volume with finite density (the Milne model illustrates this).

All such statements run up against the difficulty that we don't know what happened at early times when the universe was in the extreme conditions associated with curvature on the order of the Planck length.

Most physicists would guess that 1 is unlikely (it is sort of odd to think of a boundary where spacetime somehow stops). So we are left with 2 or 3. Both are problematic, and it is partly a matter of taste which one people feel more inclined to say is likely. I feel that to postulate an infinity of this type is too extreme a step, so I am inclined to go for 2. However, in order for a spatially flat universe to be unbounded it would have to have a non-trivial global topology, such as a 3-torus, and this is also a somewhat odd thing to propose. However, if I had to pick, then I would say this is a somewhat more palatable option than to conjecture that there is an infinite spatial region with an infinite number of galaxies etc. And it would be a conjecture. There is no way one could know that.

But fortunately I don't have to pick, because the idea of perfect spatial flatness is also somewhat unphysical, as I already said.

If there was some compelling theoretical argument to say that the average curvature $$K$$ of the universe must be exactly zero, then I would be willing to hear it. Physics at the moment does not suggest there is such an argument, so we fall back on empirical observations. Empirical work to date gives $$K = 0 \pm \epsilon$$ where $$\epsilon$$ is the experimental uncertainty. Thus the empirical work is consistent with both signs of curvature or none. We now bring in Occam's razor (the principle that one should avoid superfluous hypotheses) but this is somewhat subjective. It seems to me that to bring in the idea that there are an infinite number of galaxies is a superfluous hypothesis, when we have available the elegant and natural solution offered by the 3-sphere. In this I am influenced by the consideration that infinity is qualitatively different from any finite number. I think it is hard to support that notion that even $$1000^{1000^{1000}}$$ (or whatever) galaxies is but an infinitesimal fraction of the total, unless one has some argument to show why this is not a superfluous hypothesis compared to the simple 3-sphere.

• So you think the universe has positive curvature so forms sort of buble? – Pan Mrož Jul 27 '19 at 22:40
• In an infinite universe, however far you travel, it is always possible to go farther. Am I the only one who would find that to be more palatable than reaching a limit or returning to where you started? – D. Halsey Jul 28 '19 at 0:09
• @D.Halsey I added a paragraph to my answer in order to respond to this comment. – Andrew Steane Jul 28 '19 at 9:56
• @PanMrož yes I think that more likely than other options at the moment, but I don't know. I added a paragraph to my answer to comment on this. – Andrew Steane Jul 28 '19 at 9:57
• @AndrewSteane But doesn´t 3-sphere topology of universe need there to be more than 3+1 dimensions? Like at least 4 spatial dimensions + 1 time – Pan Mrož Jul 28 '19 at 10:16

I mean the universe couldn´t start finite time ago, expanded with finite speed and yet be infinite, that is a contradiction, so what gives?

In spatially infinite solutions of the Einstein field equations, space is infinite at all times. The big bang singularity happens at all points in this infinite space, not just at one point. Cosmological models based on general relativity do not have a central point.

Did someone tell you the universe is infinite?
We do not know as the answer lies beyond the scope of the "observable universe". Extrapolation of big bang theory makes a reasonable argument for a finite universe but there is no way to test it.

No physical infinities have been proven to exist in nature; for example, the existence of singularities is but one theory. That supports a suggestion that infinity is no more than a useful mathematical construct.

If the universe is finite, though, how is it bounded? This is not observable to us.

• This answer isn't physics, it's philosophy. – user4552 Jul 28 '19 at 17:36
• It's also wrong in detail, isn't it, if relativistic big bang theory (aka our best guess) assumes a spatially infinite universe, as you quote in your answer. As for philosophy, scientific method is one, and should be in scope; although dull to quote all the time, I thought its application might help balance the answer. Some theories are better supported than others. – JMLCarter Jul 29 '19 at 21:27

While I have never studied relativity deeply enough to properly answer this question, I still would like to say that these type of global questions always remind me of Arthur C. Clarke:

Two possibilities exist: either we are alone in the Universe or we are not. Both are equally terrifying.

I like to extrapolate this quote as:

Two possibilities exist: either there's something beyond life, or there's not. Both are equally terrifying.

And finally:

Two possibilities exist: either the universe if infinite (in time and space), or it is finite. Both are equally weird! Maybe terrifying?

@Ben Crowell above said it is a philosophical question. Indeed, the true global answer might be metaphysical. But even if the equations or experiments were to give us an answer... thing about it. Is any of the answers less disturbing? What does it mean to have an infinite universe, one that never ends and we can always move further and further? Infinite is weird! On the other hand, if there's a boundary, what lies beyond it? A periodic case seem less contradictory, but weird anyway.

My point is, any possible answer, raises more questions.